Page 130 - DMTH503_TOPOLOGY

Topology Richa Nandra, Lovely Professional University

Notes Unit 13: Compact Spaces and Compact Subspace
of Real Line

CONTENTS
Objectives

Introduction
13.1 Compact Spaces

13.2 Compact Subspaces of the Real Line
13.3 Summary

13.4 Keywords

13.5 Review Questions
13.6 Further Readings

Objectives

After studying this unit, you will be able to:
 Define open covering of a topological space;
 Understand the definition of a compact space;
 Solve the problems on compact spaces and compact subspace on real line.

Introduction

The notion of compactness is not nearly so natural as that of connectedness. From the beginning
of topology, it was clear that the closed interval [a, b] of the real line had a certain property that
was crucial for proving such theorems as the maximum value theorem and the uniform continuity
theorem. But for a long time, it was not clear how this property should be formulated for an
arbitrary topological space. It used to be thought that the crucial property of [a, b] was the fact
that every infinite subset of [a, b] has a limit point, and this property was the one dignified with
the name of compactness. Later, mathematicians realized that this formulation does not lie at
the heart of the matter, but rather that a stranger formulation, in terms of open coverings of the
space, is more central. The latter formulation is what we now call compactness. It is not as
natural of intuitive as the former; some familiarity with it is needed before its usefulness
becomes apparent.

13.1 Compact Spaces

Definition: A collection  of subsets of a space X is said to cover X, or to be a covering of X, if the
union of the elements of A is equal to X. It is called an open covering of X if its elements are open
subsets of X.

Definition: A space X is said to be compact if every open covering  of X contains a finite
sub-collection that also covers X.

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